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Properties

Label 2-750-75.17-c1-0-28
Degree $2$
Conductor $750$
Sign $-0.998 - 0.0527i$
Analytic cond. $5.98878$
Root an. cond. $2.44719$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.891 − 0.453i)2-s + (0.631 − 1.61i)3-s + (0.587 + 0.809i)4-s + (−1.29 + 1.15i)6-s + (−2.97 − 2.97i)7-s + (−0.156 − 0.987i)8-s + (−2.20 − 2.03i)9-s + (4.73 − 1.53i)11-s + (1.67 − 0.437i)12-s + (−0.801 − 1.57i)13-s + (1.30 + 4.00i)14-s + (−0.309 + 0.951i)16-s + (1.40 − 0.223i)17-s + (1.03 + 2.81i)18-s + (−1.09 + 1.50i)19-s + ⋯
L(s)  = 1  + (−0.630 − 0.321i)2-s + (0.364 − 0.931i)3-s + (0.293 + 0.404i)4-s + (−0.528 + 0.469i)6-s + (−1.12 − 1.12i)7-s + (−0.0553 − 0.349i)8-s + (−0.734 − 0.678i)9-s + (1.42 − 0.464i)11-s + (0.483 − 0.126i)12-s + (−0.222 − 0.436i)13-s + (0.347 + 1.07i)14-s + (−0.0772 + 0.237i)16-s + (0.341 − 0.0541i)17-s + (0.244 + 0.663i)18-s + (−0.250 + 0.345i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $-0.998 - 0.0527i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ -0.998 - 0.0527i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0213960 + 0.811320i\)
\(L(\frac12)\) \(\approx\) \(0.0213960 + 0.811320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 + 0.453i)T \)
3 \( 1 + (-0.631 + 1.61i)T \)
5 \( 1 \)
good7 \( 1 + (2.97 + 2.97i)T + 7iT^{2} \)
11 \( 1 + (-4.73 + 1.53i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.801 + 1.57i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-1.40 + 0.223i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (1.09 - 1.50i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.951 - 1.86i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (4.57 - 3.32i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (5.23 + 3.80i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.29 - 1.16i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (3.58 + 1.16i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-6.26 + 6.26i)T - 43iT^{2} \)
47 \( 1 + (0.732 - 4.62i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (5.05 + 0.801i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-3.44 + 10.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.99 + 9.23i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.01 + 6.39i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (-8.85 - 12.1i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (8.23 + 4.19i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (2.20 + 3.03i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.63 - 16.6i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (0.351 + 1.08i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.47 - 0.549i)T + (92.2 + 29.9i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.681238327132773955749918145909, −9.219681935910824961001211980720, −8.171898438312768119966755150960, −7.29606186979360225108148289206, −6.72253584190472837763226204177, −5.84967466605470833081935376445, −3.79617449859040098585223202412, −3.30852304191307518298080110844, −1.70981748442421873022233851224, −0.47769045413478016723360881303, 2.08988408832318053210604015019, 3.28309815981064471645880991242, 4.37426070903752636743886873512, 5.63188320320120294540226517984, 6.37936686333416541758880767586, 7.35654076028721549432823546205, 8.678101800853664879610046124488, 9.141667766756809769665691108281, 9.626812884544080809358858100789, 10.44492178014799119942209961203

Graph of the $Z$-function along the critical line